No one shall expel us from the paradise which Cantor has created for us.

David Hilbert, On the Infinite, 1926

Infinity has been puzzling humans for millennia. From mathematics to physics and philosophy, trying to understand this elusive concept has blown the minds of some of the smartest people on Earth. What is it that makes infinity so weird and mind-breaking? David Hilbert (1862-1943), the godfather of twentieth Century mathematics, came up with a story to make you feel for yourself how infinity can twist our finite minds. Let me invite you on a trip into Hilbert’s Grand Hotel, The Paradise, a place where the impossible becomes possible. If you’re afraid for those who lost their mind thinking about this, don’t worry, the asylum bill is on me (all the solutions are provided at the end of the story).

Hilbert’s Hotel

Once upon a time, in Aleph Town, there was a guy named Georg Cantor who worked as the night manager of Mr. Hilbert’s very singular hotel: The Paradise, an infinite hotel with an infinite number of rooms. Our story is set during a night when the hotel was completely full: an infinite amount of guests is peacefully sleeping in each one of the infinitely many rooms. However, during Georg’s night shift, a person knocks at the door, desperate for a room as the last train just left. Now, Mr. Hilbert takes hospitality very seriously and he’d rather fire Georg than leave a poor soul out in the cold. Our hero knows he must find a way to make room for this stranded person. Given that the hotel is full, how would you help Georg do this?

If this was not enough, Mr. Zermelo, a notoriously jolly fellow, just threw his wildest party in town, so a finite amount of people shows up at The Paradise seeking a warm place to wear off their hangovers. Mr. Hilbert is old, but once he was young too, so he’s quite sympathetic to Mr. Zermelo’s guests and chasing away these poor souls is not an option. Georg has to find a way to make room for them as well. How would you help him?

Unfortunately, there’s no time to catch breath. Mr. Fraenkel came all the way from Beth Town riding his invention: an infinite bus. He thought it was a good idea to bring his infinite amount of friends to visit Aleph Town and they all want to stay at The Paradise. It isn’t even worth mentioning that Mr. Hilbert and Mr. Fraenkel are close friends. Not being able to accommodate an infinite amount of guests would mean bankruptcy, as the hotel would lose an infinite amount of money. How should Georg face this stressful situation?

Then, as the night seems to finally calm down, the impossible becomes possible. Johnny von Neumann found a way to capitalize on Mr. Fraenkel’s invention: an infinite amount of infinite buses shows up. Losing all these guests to the competitors is not an option. Mr. Hilbert would rather sell The Paradise than see his rivals win the race for the best hotel in town. How should Georg solve this seemingly impossible riddle?

The Solutions

If you read all the way and your head didn’t explode, good for you! If you also have the solution to Georg’s problems, your math teachers should be very proud of you. If not, then rest assured, Georg still has his job. As a matter of fact, Georg is a true legend in Aleph Town, and the real reason behind its being one of the best tourist destinations in the world (you know, infinite amounts of fun). This is how he survived the night.

The first guest was the easiest to accommodate. Georg, who’s such a nice guy, kindly asked the guest in room 1 to move to room 2, guest in room 2 to move to room 3 and so on. Every guest in room n was asked to move to room n + 1. Since The Paradise is infinite there is no last room, so nobody is left out in the cold.

The second issue was just a generalization of the first. Every guest in room n was asked to move to room n + m, where m is the number of people coming from Mr. Zermelo’s party (say 3 in the figure below).

The solution to Mr. Fraenkel’s vacation was a bit more complex, but nothing terrible. Georg simply asked guests in room 1 to move to room 2, room 2 to room 4, room 3 to room 6, and so on. That is, he asked all guests in room n to move to room 2n, filling all even rooms and freeing all odd rooms. Since both even and odd numbers are infinite, the infinite amount of guests can easily fit into The Paradise.

The last situation is the hardest, but, remember, Georg is a legend. For once, he paid attention during his math class, so he knows that there is an infinite amount of prime numbers (those numbers that can be divided only by 1 and by themselves, like 2, 3, 5, 7 and so on). First, he asked every one in the hotel to move from room n to room 2n: room 3 moves to room 23 = 8, room 4 moves to room 24 = 16, and so on. Then, he took the first infinite bus and assigned each person to room 3n, where n is the number of their seat on the bus. He did so infinitely many times, assigning to each bus a prime number m and then assigning each guest the room mn, where n is the number of the guest’s seat. Since prime numbers can be divided just by 1 and by themselves, this assignment, though a bit contrived, is unique: there are no overlapping rooms (although some rooms are left empty, like room 10, which is not the power of any prime).

So, all’s well that ends well. Georg not only managed to fit more finite guests into an infinite hotel that was full, but also infinite amounts of people and even an infinite amount of infinite people! And I better not tell you of that time when Georg managed to accommodate all the guests on infinite buses coming on an infinite amount of infinite ferries across Lake Omega. Pretty wild for a night manager, isn’t it? Especially given that Georg’s salary is finite.

How big is the infinite?

Although this story is fictional, the characters are not. Georg Cantor is the father of set theory, a subject at the intersection of mathematics and philosophy (later refined by Zermelo, Fraenkel, von Neumann, and other mathematicians) through which we can grasp and manipulate the infinite. The reason why it is so important is that it provided mathematics with a solid foundation, hence why Hilbert called it a “paradise”. That is, it turns out that almost all the things mathematics bored you with during school years are sets and so there is a sense in which mathematics is “happening” within set theory. Before hating on Georg for all those afternoons on textbooks, though, consider that set theory also had some more pleasant consequences, like inspiring the development of informatics.

To explore the details a bit more, set theory answers a very simple yet fundamental question: can we measure the size of infinite sets? Cantor’s mind-blowing answer is that not only we can tell how big the infinite is but also that there are several sizes of infinity, whose comparison is not as we may expect.

Take the infinite set containing the natural numbers {0, 1, 2, 3, 4, 5, …}. It seems fair to say that this set splits up into odd {1, 3, 5, …} and even {0, 2, 4, …} numbers and that, since parts are smaller than the whole, there should be more natural numbers than there are even or odd numbers. Well, it turns out that this is not so, since one can prove that there are as many odd and even numbers as there are natural numbers (just think about the solution for the single infinite bus case).

Next take the set of rational numbers, those numbers that can be expressed as fractions (1/2, 4/3, …). Allegedly, there should be more of these than naturals, just because every natural is expressible as a base-1 fraction (1=1/1, 2=2/1 and so on…) and so the naturals are just a small part of the rationals. At this point you should be used to surprises. There actually are as many rationals as there are naturals! All these infinite sets are called “countable” meaning that we can literally count them by finding a correspondence with the set of natural numbers.

What about the different sizes of infinity? Take the set of real numbers, namely the collection which includes those numbers that cannot be represented as fractions (so-called irrationals) such as π, √2, -√3 and so on. One of Cantor’s groundbreaking results is the theorem bearing his name, stating that there are more reals than naturals: real numbers are the hallmark of “uncountable” infinity. The method of “diagonalization” used by Cantor to prove his theorem is the same that, decades later, Alan Turing would use to state the Halting Problem, which plays a crucial role in the definition of a computer. Coming back to the hotel, all the above tricks worked because we were dealing with countable amounts of incoming guests. Had an infinite bus with seat numbers π, √2, -√3 and so on arrived, then our smart night manager would have been forced to leave many of the guests out in the cold as The Paradise can only accommodate countably many guests.

Closing Thoughts

Like all mathematicians Cantor couldn’t help but use fancy words for everything (after all, The Paradise is an ∞-star hotel). So, he called the numbers that measure the size of a set “cardinal numbers” (for example, the cardinal number of the set of Italian regions is 20). Then the question becomes: can we assign cardinal numbers to infinite sets? Thanks to our brilliant night manager, we can: “aleph” and “beth” numbers are examples of infinite cardinal numbers (from the first two letters of the hebraic alphabet ‘ℵ’ and ‘ℶ’). More precisely, ℵ0 is the first infinite cardinal and marks the sizes of all countably infinite sets. After that, from ℵ1 onwards, uncountable cardinals come in greater and greater sizes!

If you are wondering how many infinite cardinals there are, well, the answer is infinitely many! Just as there are infinite many finite numbers, so there are infinitely many infinite numbers. Can we measure how many of these there are as we did above? Unfortunately not, for the same reasons that led to Russell’s Paradox, about which you can read in a previous article here on the Pamphlet. More precisely, while Russell’s Paradox tells us that there can be no set of all sets, Cantor’s Paradox tells us that there can be no set of all cardinal numbers. If there was, we could measure this set and assign a new cardinal number to it, but all cardinals were supposed to be in there already. This is yet another example of the puzzles that infinity poses to us and that are still under philosophical scrutiny nowadays.

After all these years, cardinal numbers and the size of the infinite are still the focus of contemporary research. On the one hand, set theorists are searching for super big cardinal numbers, called large cardinals. These are so big that one cannot even prove they exist, but must assume them directly as new axioms. On the other hand, philosophers of set theory try to understand this search from different perspectives. Methodologically, they pose the question of what processes lead and inform the selection and acceptance of these very large cardinals. From a foundational standpoint, the focus is currently to understand how these numbers shape our view of the universe of sets. In particular, depending on how many large cardinals we accept (or, better, which kind), some scholars say, the universe might be shaped either by order or by chaos.

Infinity is a philosophically fascinating concept. As soon as we, finite creatures, enter its realm, our most basic intuitions get blown away. Georg Cantor, however, showed us how to navigate this intricate “labyrinth of thought” without getting lost in its paradoxes. Pretty wild for a night manager!

Further Resources:

TED-Ed on Hilbert’s Hotel: https://www.youtube.com/watch?v=Uj3_KqkI9Zo
Cantor’s Attic: https://neugierde.github.io/cantors-attic/
Order vs. Caos: www.quantamagazine.org/is-mathematics-mostly-chaos-or-mostly-order-20250620/

How big is the infinite? Big… but it can be bigger

Hilbert’s Hotel

The Solutions

How big is the infinite?

Closing Thoughts

Further Resources:

Further Readings:

How big is the infinite? Big… but it can be bigger

Hilbert’s Hotel

The Solutions

How big is the infinite?

Closing Thoughts

Further Resources:

Further Readings:

Valuing Philosophy Differently